Isle Royale National Park is a 210-square-mile archipelago composed of a single large island and many small islands in Lake Superior. Moose, arrived around 1900 and by 1930, their population approached 3000, ravaging vegetation. In 1949, wolves crossed an ice bridge from Ontario. Since the late 1950s, the numbers of the moose and wolves have been tracked. (Dash indicates no data.)

(a) Integrate the Lotka-Volterra equations from 1960 through 2020. Determine the coefficient values that yield an optimal fit. Compare your simulation with the data using a time-series approach, and comment on the results.

(b) Plot the simulation of (a), but use a state-space approach.

(c) After 1993, suppose that the wildlife managers trap one wolf per year and transport it off the island. Predict how the populations of both the wolves and moose would evolve to the year 2020. Present your results as both time-series and state-space plots. For this case, as well as for (d), use the following coefficients: α = 0.3, b = 0.01111, c = 0.2l06, d = 0.0002632.

(d) Suppose that in 1993, some poachers snuck onto the island and killed 50% of the moose. Predict how the populations of both the wolves and moose would evolve to the year 2020. Present your results as both time-series and state-apace plots.

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Wolves Wolves Year Moose Year Moose 1972 23 24 700 836 22 1960 1973 802 22 1961 1974 1975 815 30 23 1962 41 1963 778 20 641 43 1976 1977 25 1964 · 507 543 33 1965 28 m ne 1978 40 1966 1967 881 24 42 22 1979 675 1980 1981 1982 50 1000 1150 966 577 22 1968 570 590 811 30 13 17 1969 1970 18 23 20 1983. 674 1971